Theorem occon 28146

Description: Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)

Assertion

Ref	Expression
occon	|- ( ( A C_ ~H /\ B C_ ~H ) -> ( A C_ B -> ( _|_ ` B ) C_ ( _|_ ` A ) ) )

Proof of Theorem occon

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssralv 3666	. . . . . 6 |- ( A C_ B -> ( A. y e. B ( x .ih y ) = 0 -> A. y e. A ( x .ih y ) = 0 ) )
2	1	ralrimivw 2967	. . . . 5 |- ( A C_ B -> A. x e. ~H ( A. y e. B ( x .ih y ) = 0 -> A. y e. A ( x .ih y ) = 0 ) )
3		ss2rab 3678	. . . . 5 |- ( { x e. ~H | A. y e. B ( x .ih y ) = 0 } C_ { x e. ~H | A. y e. A ( x .ih y ) = 0 } <-> A. x e. ~H ( A. y e. B ( x .ih y ) = 0 -> A. y e. A ( x .ih y ) = 0 ) )
4	2, 3	sylibr 224	. . . 4 |- ( A C_ B -> { x e. ~H | A. y e. B ( x .ih y ) = 0 } C_ { x e. ~H | A. y e. A ( x .ih y ) = 0 } )
5	4	adantl 482	. . 3 |- ( ( ( A C_ ~H /\ B C_ ~H ) /\ A C_ B ) -> { x e. ~H | A. y e. B ( x .ih y ) = 0 } C_ { x e. ~H | A. y e. A ( x .ih y ) = 0 } )
6		ocval 28139	. . . 4 |- ( B C_ ~H -> ( _|_ ` B ) = { x e. ~H | A. y e. B ( x .ih y ) = 0 } )
7	6	ad2antlr 763	. . 3 |- ( ( ( A C_ ~H /\ B C_ ~H ) /\ A C_ B ) -> ( _|_ ` B ) = { x e. ~H | A. y e. B ( x .ih y ) = 0 } )
8		ocval 28139	. . . 4 |- ( A C_ ~H -> ( _|_ ` A ) = { x e. ~H | A. y e. A ( x .ih y ) = 0 } )
9	8	ad2antrr 762	. . 3 |- ( ( ( A C_ ~H /\ B C_ ~H ) /\ A C_ B ) -> ( _|_ ` A ) = { x e. ~H | A. y e. A ( x .ih y ) = 0 } )
10	5, 7, 9	3sstr4d 3648	. 2 |- ( ( ( A C_ ~H /\ B C_ ~H ) /\ A C_ B ) -> ( _|_ ` B ) C_ ( _|_ ` A ) )
11	10	ex 450	1 |- ( ( A C_ ~H /\ B C_ ~H ) -> ( A C_ B -> ( _|_ ` B ) C_ ( _|_ ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   A.wral 2912   {crab 2916   C_ wss 3574   ` cfv 5888  (class class class)co 6650   0cc0 9936   ~Hchil 27776   .ih csp 27779   _|_cort 27787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-oc 28109

This theorem is referenced by:  occon2  28147  occon3  28156  ococin  28267  ssjo  28306  chsscon3i  28320  shjshsi  28351